Recently, while studying the dodecahedron, I came across this curious identity:
2*arctan(phi) = pi - arctan(2)
where arctan is the arctangent function, phi is the golden ratio, and pi is ... well, pi. :-)
As far as I can tell from brute-force calculation, the two sides of the equation are equal; however, I cannot find any algebraic justification for it. Can someone help me prove this algebraically?
(For the curious, the value of each side is the dihedral angle of the dodecahedron.)